# SL(2,3) in SL(2,5)

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) special linear group:SL(2,3) and the group is (up to isomorphism) special linear group:SL(2,5) (see subgroup structure of special linear group:SL(2,5)).
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## Definition

The group $G$ is taken as special linear group:SL(2,5): the special linear group of degree two over field:F5.

The subgroup $H$ is taken as a representative of the unique conjugacy class of subgroups of order 24. (The conjugacy class contains 5 subgroups).

This subgroup can be obtained explicitly using the quaternionic representation of special linear group:SL(2,3) over field:F5 (see also linear representation theory of special linear group:SL(2,3)).

## Arithmetic functions

Function Value Explanation
order of the whole group 120 $5^3 - 5 = 120$. See special linear group:SL(2,5) for more.
order of the subgroup 24 $3^3 - 3 = 24$. See special linear group:SL(2,3) for more.
index of the subgroup 5
size of conjugacy class of subgroup 5
number of conjugacy classes in automorphism class 1